View source for Harmonic space

A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h0465501.png" /> with a (linear) sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h0465502.png" /> of continuous real-valued functions in which three fundamental properties of classical harmonic functions (cf. [[Harmonic function|Harmonic function]]) are axiomatically fixed. These are: convergence as expressed by the second [[Harnack theorem|Harnack theorem]]; the maximum/minimum principle; and the solvability of the [[Dirichlet problem|Dirichlet problem]] for a sufficiently broad class of open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h0465503.png" />. The functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h0465504.png" /> are called harmonic functions. The advantage of this axiomatic approach consists in the fact that if it is adopted, the theory comprises not only solutions of the [[Laplace equation|Laplace equation]], but also of certain other equations of elliptic and parabolic type. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h0465505.png" /> be a locally compact topological space. A sheaf of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h0465506.png" /> is understood to be a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h0465507.png" /> defined on the family of all open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h0465508.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h0465509.png" /> such that: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655010.png" /> is a family of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655011.png" />; 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655012.png" />, then the restriction of any function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655013.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655014.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655015.png" />; and 3) for any family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655016.png" />, a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655017.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655018.png" /> if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655019.png" /> its restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655020.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655021.png" />. A sheaf of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655022.png" /> is called hyperharmonic if, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655024.png" /> is a convex cone of lower semi-continuous finite numerical functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655025.png" />. A sheaf of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655026.png" /> is said to be harmonic if, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655028.png" /> is a real vector space of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655029.png" />; the harmonic sheaf

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is used in what follows.

A locally compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655031.png" /> is called a harmonic space if the following axioms are satisfied [[#References|[3]]].

The positivity axiom: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655032.png" /> is non-degenerate at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655033.png" />, i.e. for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655034.png" /> there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655035.png" /> defined in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655036.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655037.png" />.

The convergence axiom: If an increasing sequence of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655038.png" /> is locally bounded, then it converges towards a function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655039.png" />.

The resolutivity axiom: There exists a basis of resolutive open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655040.png" />, i.e. sets such that for any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655041.png" /> of compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655042.png" /> there exists a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655043.png" /> of the Dirichlet problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655044.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655045.png" />, understood in the generalized sense of Wiener–Perron (cf. [[Perron method|Perron method]]).

The axiom of completeness: If a lower semi-continuous function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655046.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655047.png" />, which is lower finite, satisfies the condition

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on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655049.png" />, for any relatively compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655050.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655051.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655052.png" />.

The Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655054.png" />, with the sheaf of classical solutions of the Laplace equation or of the [[Thermal-conductance equation|thermal-conductance equation]] (heat equation) forms a harmonic space. There are several other variants of the axiomatics of harmonic spaces. Harmonic spaces are locally connected and do not contain isolated points; they have a basis of connected resolutive sets.

A hyperharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655055.png" /> on a harmonic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655056.png" /> is called superharmonic if for any relatively compact resolutive set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655057.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655058.png" /> is harmonic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655059.png" />. A positive superharmonic function for which any positive harmonic minorant is identically equal to zero is called a potential. A harmonic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655060.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655062.png" />-harmonic (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655064.png" />-harmonic) if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655065.png" /> there exists a positive superharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655066.png" /> (or, respectively, a potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655067.png" />) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655068.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655069.png" />.

Any harmonic space can be covered by open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655070.png" /> which satisfy the minimum principle in the following form: If a hyperharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655071.png" /> is positive outside the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655072.png" /> with any compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655073.png" /> and if

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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655075.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655076.png" />. In the case of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655077.png" />-harmonic space this minimum principle is satisfied for all open sets. The Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655078.png" /> with the sheaf of classical solutions of the Laplace equation is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655079.png" />-harmonic space if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655080.png" />, and is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655081.png" />-harmonic space if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655082.png" />; the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655084.png" />, with the sheaf of solutions of the heat equation is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655085.png" />-harmonic space.

The principal problems of the theory of harmonic spaces include the theory of solvability of the Dirichlet problem, including the behaviour of the generalized solution of this problem at boundary points. The theory of the [[Capacity|capacity]] of a set in a harmonic space, problems of balayage (cf. [[Balayage method|Balayage method]]) and the [[Robin problem|Robin problem]] have been studied.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Brélot,   "Lectures on potential theory" , Tata Inst.  (1960)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Bauer,   "Harmonische Räume und ihre Potentialtheorie" , Springer  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Constantinescu,   A. Cornea,   "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Brelot,   "On topologies and boundaries in potential theory" , Springer  (1971)</TD></TR></table>



====Comments====
Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655086.png" />-harmonic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655087.png" /> with a countable base and for which the function 1 is superharmonic admits the construction of a suitable [[Markov process|Markov process]], such that potential-theoretic notions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046550/h04655088.png" /> correspond to potential-theoretic notions of the process.

See also [[Potential theory|Potential theory]]; [[Potential theory, abstract|Potential theory, abstract]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bauer,   "Harmonische Räume" , ''Jahrbuch Ueberblicke Mathematik'' , B.I. Wissenschaftsverlag Mannheim  (1981)  pp. 9–35</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Brelot (ed.)  H. Bauer (ed.)  J.-M. Bony (ed.)  J. Deny (ed.)  G. Mokobodzki (ed.) , ''Potential theory (CIME, Stresa, 1969)'' , Cremonese  (1970)</TD></TR></table>